Local dynamics of a randomly pinned crack front during creep and forced propagation: an experimental study

Crack Propagation by Activated Avalanches during Creep and Fatigue from Elastic Interface Theory

The growth of cracks combines materials science, fracture mechanics, and statistical physics. The importance of fluctuations in the crack velocity is fundamental since it signals that the crack overcomes local barriers such as tough spots by avalanches. In ductile materials the omnipresent plasticity close to the crack tip influences the growth by history effects, which we here study in polymethylmetacrylate by various fatigue and creep protocols. We show how the crack tip local history may be encompassed in a time- and protocol-dependent length scale, which allows us to apply a statistical fracture description to the time-dependent crack growth rate, resolving the well-known paradox why fatigue cracks grow faster if the stress during a cycle is let to relax more from the peak value. The results open up novel directions for understanding fracture by statistical physics.

Scaling theory for the statistics of slip at frictional interfaces

Slip at a frictional interface occurs via intermittent events. Understanding how these events are nucleated, can propagate, or stop spontaneously remains a challenge, central to earthquake science and tribology. In the absence of disorder, rate-and-state approaches predict a diverging nucleation length at some stress σ^{*}, beyond which cracks can propagate. Here we argue for a flat interface that disorder is a relevant perturbation to this description. We justify why the distribution of slip contains two parts: a power law corresponding to "avalanches" and a "narrow" distribution of system-spanning "fracture" events. We derive novel scaling relations for avalanches, including a relation between the stress drop and the spatial extension of a slip event. We compute the cut-off length beyond which avalanches cannot be stopped by disorder, leading to a system-spanning fracture, and successfully test these predictions in a minimal model of frictional interfaces.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Effect of thresholding on avalanches and their clustering for interfaces with long-range elasticity

Avalanches are often defined as signals higher than some detection level in bursty systems. The choice of the detection threshold affects the number of avalanches, but it can also affect their temporal correlations. We simulated the depinning of a long-range elastic interface and applied different thresholds including a zero one on the data to see how the sizes and durations of events change and how this affects temporal avalanche clustering. Higher thresholds result in steeper size and duration distributions and cause the avalanches to cluster temporally. Using methods from seismology, the frequency of the events in the clusters was found to decrease as a power-law of time, and the size of an event in a cluster was found to help predict how many events it is followed by. The results bring closer theoretical studies of this class of models to real experiments, but also highlight how different phenomena can be obtained from the same set of data.

Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces

We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.

Thermal dissipation as both the strength and weakness of matter. A material failure prediction by monitoring creep

In any domain involving some stressed solids, that is, from seismology to general engineering, the strength of matter is a paramount feature to understand. We here discuss the ability of a simple thermally activated sub-critical model, which includes the auto-induced thermal evolution of cracks tips, to predict the catastrophic failure of a vast range of materials. It is in particular shown that the intrinsic surface energy barrier, for breaking the atomic bonds of many solids, can be easily deduced from the slow creeping dynamics of a crack. This intrinsic barrier is however higher than the macroscopic load threshold at which brittle matter brutally fails, possibly as a result of thermal activation and of a thermal weakening mechanism. We propose a novel method to compute this macroscopic energy release rate of rupture, Ga, solely from monitoring slow creep, and we show that this reproduces the experimental values within 50% accuracy over twenty different materials, and over more than four decades of fracture energy.

Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions

Disordered elastic interfaces display avalanche dynamics at the depinning transition. For short-range interactions, avalanches correspond to compact reorganizations of the interface well described by the depinning theory. For long-range elasticity, an avalanche is a collection of spatially disconnected clusters. In this Letter we determine the scaling properties of the clusters and relate them to the roughness exponent of the interface. The key observation of our analysis is the identification of a Bienaymé-Galton-Watson process describing the statistics of the number of clusters. Our work has concrete importance for experimental applications where the cluster statistics is a key probe of avalanche dynamics.

How heat controls fracture: the thermodynamics of creeping and avalanching cracks

While of paramount importance in material science, the dynamics of cracks still lacks a complete physical explanation. The transition from their slow creep behavior to a fast propagation regime is a notable key, as it leads to full material failure if the size of a fast avalanche reaches that of the system. We here show that a simple thermodynamics approach can actually account for such complex crack dynamics, and in particular for the non-monotonic force-velocity curves commonly observed in mechanical tests on various materials. We consider a thermally activated failure process that is coupled with the production and the diffusion of heat at the fracture tip. In this framework, the rise in temperature only affects the sub-critical crack dynamics and not the mechanical properties of the material. We show that this description can quantitatively reproduce the rupture of two different polymeric materials (namely, the mode I opening of polymethylmethacrylate (PMMA) plates, and the peeling of pressure sensitive adhesive (PSA) tapes), from the very slow to the very fast fracturing regimes, over seven to nine decades of crack propagation velocities. In particular, the fastest regime is obtained with an increase of temperature of thousands of Kelvins, on the molecular scale around the crack tip. Although surprising, such an extreme temperature is actually consistent with different experimental observations that accompany the fast propagation of cracks, namely, fractoluminescence (i.e., the emission of visible light during rupture) and a complex morphology of post-mortem fracture surfaces, which could be due to the sublimation of bubbles.

Universal Scaling of the Velocity Field in Crack Front Propagation

The propagation of a crack front in disordered materials is jerky and characterized by bursts of activity, called avalanches. These phenomena are the manifestation of an out-of-equilibrium phase transition originated by the disorder. As a result avalanches display universal scalings which are, however, difficult to characterize in experiments at a finite drive. Here, we show that the correlation functions of the velocity field along the front allow us to extract the critical exponents of the transition and to identify the universality class of the system. We employ these correlations to characterize the universal behavior of the transition in simulations and in an experiment of crack propagation. This analysis is robust, efficient, and can be extended to all systems displaying avalanche dynamics.

Depinning Dynamics of Crack Fronts

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatiotemporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value v_{m} can be predicted from an overdamped equation of motion (v-v_{m})/v_{0}=[G-G_{c}(v_{m})]/G_{c}(v_{m}) involving the characteristic material speed v_{0}=G_{c}(v_{m})/G_{c}^{'}(v_{m}) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.

Avalanches and extreme value statistics in interfacial crackling dynamics

We study the avalanche and extreme statistics of the global velocity of a crack front, propagating slowly along a weak heterogeneous interface of a transparent polymethyl methacrylate block. The different loading conditions used (imposed constant velocity or creep relaxation) lead to a broad range of average crack front velocities. Our high-resolution and large dataset allows one to characterize in detail the observed intermittent crackling dynamics. We specifically measure the size S, the duration D, as well as the maximum amplitude [Formula: see text] of the global avalanches, defined as bursts in the interfacial crack global velocity time series. Those quantities characterizing the crackling dynamics follow robust power-law distributions, with scaling exponents in agreement with the values predicted and obtained in numerical simulations of the critical depinning of a long-range elastic string, slowly driven in a random medium. Nevertheless, our experimental results also set the limit of such model which cannot reproduce the power-law distribution of the maximum amplitudes of avalanches of a given duration reminiscent of the underlying fat-tail statistics of the local crack front velocities.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Thermally activated crack fronts propagating in pinning disorder: simultaneous brittle/creep behaviour depending on scale

We study theoretically the propagation of a crack front in mode I along an interface in a disordered elastic medium, with a numerical model considering a thermally activated rheology, toughness disorder and long-range elastic interactions. This model reproduces not only the large-scale dynamics of the crack front position in fast or creep loading regimes, but also the small-scale self-affine behaviour of the front. Two different scaling laws are predicted for the front morphology, with a Hurst exponent of 0.5 at small scales and a logarithmic scaling law at large scales, consistently with experiments. The prefactor of these scaling laws is expressed as a function of the temperature, and of the quenched disorder characteristics. The cross-over between these regimes is expressed as a function of the quenched disorder amplitude, and is proportional to the average energy release rate, and to the inverse of temperature. This model captures as well the experimentally observed local velocity fluctuation probability distribution, with a high-velocity tail P(v)∼v -2.6 This feature is shown to arise when the quenched disorder is sufficiently large, whereas smaller toughness fluctuations lead to a lognormal-like velocity distribution. Overall, the system is shown to obey a scaling determined by two distinct mechanisms as a function of scale: namely, the large scales display fluctuations similar to an elastic line in an annealed noise excited as the average front travels through the pinning landscape, while small scales display a balance between thresholds in possible elastic forces and quenched disorder.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Mode-III interfacial crack propagation in heterogeneous media

We monitor optically the propagation of a slow interfacial mode III crack along a heterogeneous weak interface and compare it to mode I loading. Pinning and depinning of the front on local toughness asperities within the process zone are the main mechanisms for fracture roughening. Geometrical properties of the fracture fronts are derived in the framework of self-affine scale invariance and Family-Vicsek scaling. We characterize the small and large scale roughness exponents ζ_{-}=0.6 and ζ_{+}=0.35, the growth exponent at large scale β_{+}=0.58, and the power-law exponent of the local velocity distribution of the fracture fronts, η=2.55. All these analyzed properties are similar to those previously observed for mode I interfacial fractures. We also observe a common power-law decay of the probability distribution function of avalanche area. We finally observe that amplitude of front fluctuations, local rupture velocity correlation in time, and larger size of events highlight more dynamically unstable behavior of mode III crack ruptures.

Crack propagation through disordered materials as a depinning transition: A critical test of the theory

The dynamics of a planar crack propagating within a brittle disordered material is investigated numerically. The fracture front evolution is described as the depinning of an elastic line in a random field of toughness. The relevance of this approach is critically tested through the comparison of the roughness front properties, the statistics of avalanches, and the local crack velocity distribution with experimental results. Our simulations capture the main features of the fracture front evolution as measured experimentally. However, some experimental observations such as the velocity distribution are not consistent with the behavior of an elastic line close to the depinning transition. This discrepancy suggests the presence of another failure mechanism not included in our model of brittle failure.

Interevent Correlations from Avalanches Hiding Below the Detection Threshold

Numerous systems ranging from deformation of materials to earthquakes exhibit bursty dynamics, which consist of a sequence of events with a broad event size distribution. Very often these events are observed to be temporally correlated or clustered, evidenced by power-law-distributed waiting times separating two consecutive activity bursts. We show how such interevent correlations arise simply because of a finite detection threshold, created by the limited sensitivity of the measurement apparatus, or used to subtract background activity or noise from the activity signal. Data from crack-propagation experiments and numerical simulations of a nonequilibrium crack-line model demonstrate how thresholding leads to correlated bursts of activity by separating the avalanche events into subavalanches. The resulting temporal subavalanche correlations are well described by our general scaling description of thresholding-induced correlations in crackling noise.

Bursts of activity in collective cell migration

Dense monolayers of living cells display intriguing relaxation dynamics, reminiscent of soft and glassy materials close to the jamming transition, and migrate collectively when space is available, as in wound healing or in cancer invasion. Here we show that collective cell migration occurs in bursts that are similar to those recorded in the propagation of cracks, fluid fronts in porous media, and ferromagnetic domain walls. In analogy with these systems, the distribution of activity bursts displays scaling laws that are universal in different cell types and for cells moving on different substrates. The main features of the invasion dynamics are quantitatively captured by a model of interacting active particles moving in a disordered landscape. Our results illustrate that collective motion of living cells is analogous to the corresponding dynamics in driven, but inanimate, systems.

How cracks are hot and cool: a burning issue for paper

Material failure is accompanied by important heat exchange, with extremely high temperature - thousands of degrees - reached at crack tips. Such a temperature may subsequently alter the mechanical properties of stressed solids, and finally facilitate their rupture. Thermal runaway weakening processes could indeed explain stick-slip motions and even be responsible for deep earthquakes. Therefore, to better understand catastrophic rupture events, it appears crucial to establish an accurate energy budget of fracture propagation from a clear measure of various energy dissipation sources. In this work, combining analytical calculations and numerical simulations, we directly relate the temperature field around a moving crack tip to the part α of mechanical energy converted into heat. By monitoring the slow crack growth in paper sheets using an infrared camera, we measure a significant fraction α = 12% ± 4%. Besides, we show that (self-generated) heat accumulation could weaken our samples by microfiber combustion, and lead to a fast crack/dynamic failure/regime.

Fractal frontiers of bursts and cracks in a fiber bundle model of creep rupture

We investigate the geometrical structure of breaking bursts generated during the creep rupture of heterogeneous materials. Using a fiber bundle model with localized load sharing we show that bursts are compact geometrical objects; however, their external frontiers have a fractal structure which reflects their growth dynamics. The perimeter fractal dimension of bursts proved to have the universal value 1.25 independent of the external load and of the amount of disorder in the system. We conjecture that according to their geometrical features, breaking bursts fall in the universality class of loop-erased self-avoiding random walks with perimeter fractal dimension 5/4 similar to the avalanches of Abelian sand pile models. The fractal dimension of the growing crack front along which bursts occur proved to increase from 1 to 1.25 as bursts gradually cover the entire front.

Statistical properties of Barkhausen noise in amorphous ferromagnetic films

We investigate the statistical properties of the Barkhausen noise in amorphous ferromagnetic films with thicknesses in the range between 100 and 1000 nm. From Barkhausen noise time series measured with the traditional inductive technique, we perform a wide statistical analysis and establish the scaling exponents τ,α,1/σνz, and ϑ. We also focus on the average shape of the avalanches, which gives further indications on the domain-wall dynamics. Based on experimental results, we group the amorphous films in a single universality class, characterized by scaling exponents τ=1.28±0.02,α=1.52±0.3, and 1/σνz=ϑ=1.83±0.03, values compatible with that obtained for several bulk amorphous magnetic materials. Besides, we verify that the avalanche shape depends on the universality class. By considering the theoretical models for the dynamics of a ferromagnetic domain wall driven by an external magnetic field through a disordered medium found in literature, we interpret the results and identify an experimental evidence that these amorphous films, within this thickness range, present a typical three-dimensional magnetic behavior with predominant short-range elastic interactions governing the domain-wall dynamics. Moreover, we provide experimental support for the validity of a general scaling form for the average avalanche shape for non-mean-field systems.

Adhesion failure at 180,000 frames per second: direct observation of the detachment process of a mushroom-shaped adhesive

Nature has successfully evolved the mushroom-shaped contact geometry in many organisms in order to solve the attachment problem. We studied the detachment process of individual bioinspired artificial mushroom-shaped adhesive microstructures (MSAMSs) resolving the failure dynamics at high spatiotemporal resolution. The experimental data provide strong evidence for a homogeneous stress distribution in MSAMS, which was recently proposed. Our results allow us to explain the advantage of such contact geometry and provide a suggestion for the widely observed mushroom-shaped contact geometry.

Temporal and spacial evolution of bursts in creep rupture

We investigate the temporal and spacial evolution of single bursts and their statistics emerging in heterogeneous materials under a constant external load. Based on a fiber bundle model we demonstrate that when the load redistribution is localized along a propagating crack front, the average temporal shape of pulses has a right-handed asymmetry; however, for long range interaction a symmetric shape with parabolic functional form is obtained. The pulse shape and spatial evolution of bursts proved to be correlated, which can be exploited in materials' testing. The probability distribution of the size and duration of bursts have power law behavior with a crossover to higher exponents as the load is lowered. The crossover emerges due to the competition of the slow and fast modes of local breaking being dominant at low and high loads, respectively.

Avalanche dynamics of elastic interfaces

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines wetting a nonhomogeneous substrate, or cracks in brittle disordered material proceed via intermittent motion, called avalanches. Here we develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d≥d(uc) (d(uc)=2,4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk [Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model]. Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (nonlinear) instanton equation. We focus on the limit of slow uniform driving. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder, up to the renormalization of two parameters at d=d(uc). We calculate a number of observables of direct experimental interest: Both for the center of mass, and for a given Fourier mode q, we obtain various correlations and probability distribution functions (PDF's) of the velocity inside an avalanche, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at nonzero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e., including loop corrections, the one-time PDF of the center-of-mass velocity u[over ·] for dimension d<d(uc). The singularity at small velocity P(u[over ·])~1/u[over ·](a) is substantially reduced from a=1 (MFT) to a=1-2/9(4-d)+... (short-ranged elasticity) and a=1-4/9(2-d)+... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.

Non-Gaussian nature of fracture and the survival of fat-tail exponents

We study the fluctuations of the global velocity V(l)(t), computed at various length scales l, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity--and extreme event distributions--by measuring only at a global scale.

Evolution of the average avalanche shape with the universality class

A multitude of systems ranging from the Barkhausen effect in ferromagnetic materials to plastic deformation and earthquakes respond to slow external driving by exhibiting intermittent, scale-free avalanche dynamics or crackling noise. The avalanches are power-law distributed in size, and have a typical average shape: these are the two most important signatures of avalanching systems. Here we show how the average avalanche shape evolves with the universality class of the avalanche dynamics by employing a combination of scaling theory, extensive numerical simulations and data from crack propagation experiments. It follows a simple scaling form parameterized by two numbers, the scaling exponent relating the average avalanche size to its duration and a parameter characterizing the temporal asymmetry of the avalanches. The latter reflects a broken time-reversal symmetry in the avalanche dynamics, emerging from the local nature of the interaction kernel mediating the avalanche dynamics.

Scaling laws for impact fragmentation of spherical solids

We investigate the impact fragmentation of spherical solid bodies made of heterogeneous brittle materials by means of a discrete element model. Computer simulations are carried out for four different system sizes varying the impact velocity in a broad range. We perform a finite size scaling analysis to determine the critical exponents of the damage-fragmentation phase transition and deduce scaling relations in terms of radius R and impact velocity v(0). The scaling analysis demonstrates that the exponent of the power law distributed fragment mass does not depend on the impact velocity; the apparent change of the exponent predicted by recent simulations can be attributed to the shifting cutoff and to the existence of unbreakable discrete units. Our calculations reveal that the characteristic time scale of the breakup process has a power law dependence on the impact speed and on the distance from the critical speed in the damaged and fragmented states, respectively. The total amount of damage is found to have a similar behavior, which is substantially different from the logarithmic dependence on the impact velocity observed in two dimensions.

Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium

We study the average velocity of crack fronts during stable interfacial fracture experiments in a heterogeneous quasibrittle material under constant loading rates and during long relaxation tests. The transparency of the material (polymethylmethacrylate) allows continuous tracking of the front position and relation of its evolution to the energy release rate. Despite significant velocity fluctuations at local scales, we show that a model of independent thermally activated sites successfully reproduces the large-scale behavior of the crack front for several loading conditions.